Why does estimation error increase with $|H|$ and decrease with $m$ in PAC learning?
Definitely, you can find the proof in different resources (for example, in these notes or in the paper that originally proposed PAC learnability, A Theory of the Learnable). However, the intuition behind your question is when the size of the hypothesis increases, if you do not change anything, you can't see more part of the space. Hence, the estimation error will increase. Moreover, when you increase the number of samples, you have more chance to see more part of the hypothesis space, hence, the estimation error decrease.
Also, you can see some lemma about the relation of the PAC learnability and other similar concepts in the Wikipedia article Probably approximately correct learning:

Under some regularity conditions these three conditions are equivalent:

The concept class $C$ is PAC learnable.
The VC dimension of $C$ is finite.
$C$ is a uniform Glivenko-Cantelli class.